Question

If $x. \frac{dy}{dx} + y = x. \frac{f\left(xy\right)}{f'\left(xy\right)}$, then $f\left(xy\right)$ is equal to

• A. $k.e^{\frac{x^2}{2}}$
• B. $k.e^{\frac{y^2}{2}}$
• C. $k.e^{x^2}$
• D. $k.e^{\frac{xy}{2}}$

Answer 1

Answer: (A)

Given, $x. \frac{dy}{dx} + y = x. \frac{f\left(xy\right)}{f'\left(xy\right)}$
i.e., $\frac{d}{dx} \left(xy\right) = x \frac{f\left(x,y\right)}{f'\left(x,y\right)}$
$\Rightarrow \frac{f'\left(xy\right)}{f\left(xy\right)} d\left(xy\right) = x dx$
$\Rightarrow \int \frac{f'\left(xy\right)}{f\left(xy\right)}d \left(xy\right) = \int xdx$
$\Rightarrow \log\left[f\left(xy\right)\right] = \frac{x^{2}}{2 } + C$
$\Rightarrow f\left(xy\right) = e^{\left(x^2 2+C\right)}$
$= e^{\frac{x^2}{2}} e^{C} = k. e^{\frac{x^2}{2}} $